Broken symmetry, excitons, gapless modes and topological excitations in Trilayer Quantum Hall systems Jinwu Ye Department of Physics, The Pennsylvania State University, University Park, PA, 16802 5 (February 2, 2008) 0 WestudytheinterlayercoherentincompressiblephaseinTrilayerQuantumHallsystems(TLQH) 0 2 attotalfillingfactorνT =1fromthreeapproaches: MutualCompositeFermion(MCF),Composite Boson (CB) and wavefunction approach. Just like in Bilayer Quantum Hall system, CB approach n is superior than MCF approach in studyingTLQHwith brokensymmetry. TheHall and Halldrag a resistivities are found to be quantized at h/e2. Two neutral gapless modes with linear dispersion J relations are identified and the ratio of the two velocities is close to √3. The novel excitation 5 spectra are classified into two classes: Charge neutral bosonic 2-body bound states and Charge 2 1 fermionic 3-body bound states. In general, there are two 2-body Kosterlize-Thouless (KT) ± transition temperatures and one 3-body KT transition. The Charge 1 3-body fermionic bound ] ± l states may be the main dissipation source of transport measurements. The broken symmetry in l a terms of SU(3) algebra is studied. The structure of excitons and their flowing patterns are given. h The coupling between the two Goldstone modes may lead to the broadening in the zero-bias peak - in the interlayer correlated tunnelings of the TLQH. Several interesting features unique to TLQH s e are outlined. Limitations of theCB approach are also pointed out. m . I. INTRODUCTION ture Kosterlize-Thouless (KT) phase transition at TKT t a where boundstates of the 4 flavorsof meronsarebroken m into free merons. Unfortunately, this transition has not Quantum Hall Effect (FQHE) in multicomponent sys- - been observed so far. The EPQFM approach is a micro- d tems has attracted a lot of attention since the seminal scopic one which takes care of LLL projection from the n work by Halperin [1]. These components could be the verybeginning. However,thechargesectorwasexplicitly o spinsofelectronswhentheZeemancouplingisverysmall c projected out, the connection and coupling between the or layer indices in multi-layered system. In particular, [ chargesectorwhichdisplaysFractionalQuantumHallef- spin-polarized Bilayer Quantum Hall (BLQH) systems fect and the spin sector which displays interlayer phase 7 at total filling factor ν = 1 have been under enormous v T coherence was not obvious in the EPQFM approach. experimental [2–5] and theoretical [6–13] investigations 9 overthe lastdecade. Whenthe interlayerseparationd is In [12], I used and extended both Mutual Composite 5 4 sufficiently large, the bilayer system decouples into two Fermion (MCF) and Composite Boson (CB) approaches 2 separatecompressibleν =1/2layers. However,whendis tostudybothbalancedandimbalancedBLQHandmade 0 sufficiently small, in the absence of interlayer tunneling, criticalcomparisonsbetweenthetwoapproaches. Iiden- 4 the system undergoes a quantum phase transition into tified several serious problems with the MCF approach 0 a novel spontaneous interlayer coherent incompressible and then developed a simple CB approach which natu- / t phase [11]. rally avoids all these problems suffered in the MCF ap- a m By treating the two layer indices as two pseudo- proach. I found the CB approach is superior to MCF spin indices, Girvin, Macdonaldandcollaborators[9–11] approachintheBLQHwithbrokensymmetry. Thefunc- - d mapped the bilayer system into a Easy Plane Quantum tionalformofthespinsectoroftheCBtheoryisthesame n Ferromagnet (EPQFM). They achieved the mapping by astheEPQFM.Italsohastheadvantagetotreatcharged o projectingtheHamiltonianoftheBLQHontotheLowest excitations and the collective order parameter fluctua- c : LandauLevel(LLL)andthenusingsubsequentHartree- tionsinthepseudo-spinsectoronthesamefooting,there- v Fock(HF)approximationandgradientexpansion(called fore can spell out the spin-charge connections explicitly. i X LLL+HF in the following ). They explored many rich From this spin-charge connection, we are able to classify r and interesting physical phenomena in this system. The all the possible excitations in the BLQH in a systematic a low energy excitations above the ground state is given way. The CB theory may also be applied to study the by aneffective 2+1dimensionalXY model. In addition incoherent disordered insulating side and the quantum to the Goldstone mode, there are also 4 flavors of topo- phase transitions between different ground states [13]. logical defects called ” merons ” which carry fractional UsingtheCBapproach,Ialsostudiedseveralinteresting charges 1/2 and also have vorticity. They have log- phenomena specific to im-balanced BLQH. Just like any ± ± arithmic divergentself energiesand are bound into pairs Chern-Simontheory,theCBtheoryinBLQHhasitsown at low temperature. The lowestenergy excitations carry limitations: it is hard to incorporate the LLL projection charge e which are a meron pair with opposite vor- ( see however [14] ), some parameters can only be taken ± ticity and the same charge. There is a finite tempera- asphenomilogicalparameterstobefittedintothemicro- 1 scopic LLL+HF calculations or experimental data. It is operators and normal ordered electron densities on the an effective low energy theory, so special care is needed three layers. The intralayerinteractions areV =V = 11 22 to capture some physics at microscopic length scales V = e2/ǫr, while interlayer interaction is V = V = 33 12 21 Inthis paper,we use both MCF approachandthe CB V = V = e2/ǫ√r2+d2,V = V = e2/ǫ√r2+4d2 23 32 13 31 approach developed in [12] to study spin polarized Tri- where ǫ is the dielectric constant. layer Quantum Hall (TLQH) systems at total filling fac- Therestofthepaperisorganizedasfollows. Insection tor ν = 1. We also supplement the two approaches by II,wediscuss MCFapproachtoTLQHandpointoutits T wavefunction approach. TLQH is interesting from both weakness. In section III, we discuss CB approach and experimental and theoreticalsides. On the experimental derive an effective action involving the charge and the side, TLQH systems have been fabricated in high mo- two Goldstone modes. We also derive a dual action of bility electron systems in the experimental group led by theCBapproachandmakecriticalcomparisonsbetween Shayegan[15]. Onthetheoryside,becausethethreelay- this dual action and the MCF action. We classify all ersplaythe rolesofthreeflavors. The excitonsinBLQH the possible excitations and discuss possible 2-body and is the pairing of particle in one layer and the hole in the 3-body bound states and associated Kosterlize-Thouless other. This pairing structure in BLQH is similar to the transitions. In section IV, we will outline several salient CooperpairinginBCSsuperconductorsupto aparticle- features of correlated interlayer tunnelings with or with- hole transformation. While the excitons in TLQH is not out in-plane magnetic field in TLQH. We reach conclu- obvious,becausesofarthereis noanalogofthreeflavors sions in the final section. At appropriate places in the superconductors yet. It would be interesting to under- paper, we point out the limitations of the CB approach. stand the structures of the excitons and all the possible excitationsin TLQH.As showninthis paper,due to the particularstructurepatternoftheexcitons,therearetwo II. MUTUAL COMPOSITE FERMION coupledGoldstonemodesintheTLQH(seeEqn.33). In APPROACH: thepresenceofinterlayertunnelings(withorwithoutin- plane magnetic field ), the interference between the two We can extend the single-valued singular gauge trans- gapless modes may also lead to many new phenomena formation in [12] to tri-layer system: which can not be seen in BLQH. There are at leasttwo welldefined regimes for TLQH. U =e2i αβ d2x d2x′Uαβρα(~x)arg(~x−~x′)ρβ(~x′) (2) (I) Interlayercoherentregime: whenthe distance dbew- P R R teen the two adjcent layers is sufficiently small, then all where the 3 3 symmetric matrix U and its inverse is the three layers are strongly correlated. (II) Weakly- × given by: coupled regime: when d is sufficiently large, then all the three layers are are weakly coupled. Depending on the 0 1 1 1 1 1 distance d, there could be many other possible regimes. U =1 0 1  U−1 = 1−1 1 1  (3) t t 2 − In this paper, we focus on regime (I) which is the in- 1 1 0 1 1 1    −  terlayer coherent regime. The system in the interlayer coherent regime (I) was discussed in [6,16] in LLL+ HF The Hamiltonian Eqn.1 is transformed into: approach, two Goldstone modes are found. Here, we study the regime (I) in detail from both MCF and CB H = d2xψ†(~x)(−i¯h∇~ + ecA~(~x)−¯h~aα(~x))2ψ (~x) (4) approachesand alsostress the brokensymmetry state in 0 Z α 2m α terms of SU(3) algebra. Consider a tri-layer system with N1,N2,N3 electrons where the transformed fermion ψα(~x) = Ucα(~x)U−1 is in layer1 ( the bottom layer), layer2 ( the middle layer given by: )andlayer3(thetoplayer)respectivelyinthepresence ofmagneticfieldB~ = A~. Weassumeequalinterlayer ψα(~x)=e2i d2x′Uαβ[arg(~x−~x′)+arg(~x′−~x)]ρβ(~x′)cα(~x) (5) ∇× R distance d between adjacent layers, the total filling fac- tor is νT = 1 and the spin is completely polarized. The andthethreeChern-Simon(CS)gaugefieldsaα inEqn.4 Hamiltonian H =H0+Hint is satisfies: ∇·~aα = 0,∇×~aα = 2πUαβρβ(~x) = 2π[ρ(~x)− ρ (~x)] where ρ(~x)= ρ (~x) is the totaldensity of the α α α ( i¯h~ + eA~(~x))2 system. P H = d2xc†(~x) − ∇ c c (~x) 0 Z α 2m α In the following, we put h¯ = c = e = ǫ = 1. At total fillingfactorν =1, A~ =2πnwheren=n +n +n 1 T 1 2 3 Hint = 2Z d2xd2x′δρα(~x)Vαβ(~x−~x′)δρβ(x~′) (1) isthetotalaverageel∇ect×rondensity. Byabsorbingtheav- eragevalues of CS gauge fields <~a >=2π(n n ) α α ∇× − where electrons have bare mass m and carry charge e; into the external gauge potential A~∗ = A~ <~a >, we − α − α c ,δρ (~x) = c†(~x)c (~x) ρ¯ ,α = 1,2,3 are electron have: α α α α − α 2 H0 =Z d2xψα†(~x)(−i∇~ +A~∗α(2~xm)−δ~aα(~x))2ψα(~x) (6) L= 4iqπa0catc+ 16qπ(atc)2 ǫ 1 d + cq2(a0)2+ [ǫ ω2+(χ )q2](at)2 6 c 6 c c− 3π c where A~∗ = 2πn and δ~a = 2πU δρ (~x) = 2π[δρ(~x∇)×− δραα(~x)] arαe the ∇de×viatiαons fromαβtheβcorre- + ǫ4lq2(a0l)2+ 41[ǫlω2+(χl+ πd)q2](atl)2 sponding average density. ǫ 1 d When d < dc/2, the strong intra- and inter-layer in- + 1r2q2(a0r)2+ 12[ǫrω2+(χr+ 3π)q2](atr)2 teractions renormalize the bare mass into two different δǫ 1 d effective masses m∗ = m∗,m∗ [14]. MCF in each layer + q2a0a0+ [δǫω2+(δχ+ )q2]atat + (7) 1 3 2 9 c r 9 4π c r ··· feeleffectivemagneticfieldB∗ = A~∗ =2πn ,there- fore fill exactlyone MCF Lanαdau∇lev×el.αThe eneαrgygaps where~as =~a1+~a2+~a3,~al =~a1−~a3,~ar =~a1+~a3−2~a2 whichstandforcenterofmass,leftandrightchannelsre- are simply the cyclotrongaps of the MCF Landau levels ω∗ = Bα∗. spectively[17]. ǫc =(2ǫ1+ǫ2)/3,χc =(2χ1+χ2)/3,ǫl = cα m∗α ǫ1,χl = χ1,ǫr = (ǫ1 +2ǫ2)/3,χr = (χ1 +2χ2)/3,δǫ = Transport properties: Adding three different source ǫ ǫ ,δχ=χ χ . Thedielectricconstantsǫ = m∗α ggarautgiengpooutetntthiaelsMAC~sαFtψoαtfihrestt,htrheeenlathyeermsuintuEalqCn.S6,gainutgee- asi1nn−dglet2hleayseursscyesp1tt−eimbil2iitnie[s18χ].α = a2rπe1mh∗αigwheerregrcaadlciαeunlattte2edπrBmiα∗ns fields, we can calculate the resistivity tensor by Kubo ··· and at is the transverse component of gauge fields in formula. We find that both Hall resistivity and two Hall α Coulomb gauge ~a = 0. The system is invariant drag resistivities are h/e2. This means if one drives cur- ∇ · α under the Z symmetry which exchanges layer 1 and 3, rent in one layer, the Hall resistivities in all the three 2 namely a a ,a a ,a a . This symmetry layersarequantizedath/e2,whilethelongitudinalresis- c → c l → − l r → r dictatesthatonly a a ,a a ,a a ,a a canappearinthe tivitiesinallthethreelayersvanish. Thisisthehallmark c c c r r r l l Maxwell terms. The last two terms in Eqn. 7 which of interlayer coherent quantum hall states. is the coupling between a and a can be shown to be c r In the following, we confine to balanced case N = 1 irrelevant in the low energy limit. N =N =N/3,imbalancedcasecanbediscussedalong 2 3 More intuitive choices are a = a a ,a = a a u 1 2 d 2 3 the similar line developed in [12]. − − which is related to a ,a by a = a +a ,a = a a . l r l u d r u d Fractional charges: Let’s look at the charge of quasi- But a and a are coupled, the two normal mode−s are u d particlescreatedbyMCFfieldoperatorsψ†(~x). Ifwein- a ,a instead of a ,a . As expected, there is a Chern- α l r u d sertoneelectronatoneofthelayers,saylayer1,fromthe Simon term for the center of mass gauge field ~a , while s singular gauge transformation in Eqn.2, we can see this there are two Maxwell terms for the left and right chan- is equivalent to insert one MCF in layer 1, at the same nelgaugefields~a ,~a whicharegapless. The twogapless l r time, insertoneflux quantumatlayer2andanotherone modes stand for the relative charge density fluctuations at layer 3 directly above the position where we inserted among the three layers. After putting back h¯,c,e,ǫ, we the electron. The inserted flux quantum at layer 2 and find the two spin-wave velocities in terms of experimen- layer3 areinthe oppositedirectiontothe externalmag- tally measurable parameters: neticfield,thereforepromoteoneMCFinlayer2andone 3(ω∗)2 2αc d ω∗ inlayer3tothesecondMCFLandaulevelinlayer2and v2 = c +( )( ) c l 2πn ǫ l √2πn layer3 respectively. Becauseinserting one electronleads to three MCFs in each layer, we conclude the charge of v2 = 3(ωc∗)21+2γ−1 +(2αc)(d) ωc∗ 1 (8) eachMCFis-1/3. Theaboveargumentgivesthecorrect r 2πn 1+2γ ǫ l √2πn1+2γ totalfractionalchargesofMCF,butitcannotdetermine where we set ω∗ =ω∗ and n is the total density, l is the c c1 the relative charge distributions among the three layers. magneticlength,γ istheratioofthetwoeffectivemasses At mean field level, the energies of all the possible rela- γ =m∗/m∗ andα 1/137isthefinestructureconstant. tivechargedifferencesaredegenerate. Thelowestenergy 2 1 ∼ As in BLQH [12], we expect the second term domi- configuration can only be determined by fluctuations. nates the first, then v/v √1+2γ. namely, the ratio l r ∼ Left- and right-moving gapless modes: δρα(~x) can be ofthetwovelocitiesisdeterminedbytheratioofthetwo expressed in terms of the CS gauge fields δρα(~x) = effectivemasses. Ifwetaketheeffectivemassestobethe [U−1] δ~aβ whereU−1isgiveninEqn.3. Thisconstraint band mass, then v /v √3. t αβ 2π t l r ∼ canbeimposedbyLagrangianmultipliersa0 whichplays Topological excitations: As discussed in the previous α theroleoftime componentsofCSgaugefields. Integrat- paragraph,theMCFcarriescharge1/3,buttheirrelative ingoutMCFψ ,ψ ,ψ toone-loopandcarefullyexpand- charge distributions among the three layers are undeter- 1 2 3 ing the interlayer Coulomb interaction to the necessary mined. They can be characterized by their (a ,a ,a ) c l r order in the long-wavelength limit leads to an effective charges ( 1/3,q,q ) with q,q = 0, 1, . Exchang- l r l r ± ± ··· action for the three gauge fields. ing a leads to 1/r interaction between two MCF, while c 3 exchanginga ,a leads to logarithmic interactionswhich on the same footing and explicitly stress the spin and l r mayleadtoa2-bodyboundstatebetweentwoMCFwith charge connections. We classify all the possible excita- opposite (q ,q ). The energy of this bound state with tions, bound states and associated KT transitions. l r length L is ∆ +∆ +q2e2 (q q + qr1qr2)h¯ω∗lnL/l By defining the center of mass, left and right channels + − c L − l1 l2 1+2γ c where ∆ ,∆ are the core energies of QH and QP re- densities as: + − spectively. There are also possible 3-body bound states δρ =δρ +δρ +δρ of three MCF with q = q =0. c 1 2 3 i li i ri Unfortunately,thePgluingcPonditions(orselectionrules δρ =δρ δρ l 1 3 − ) of (qc,ql,qr) for realizable physical excitations are not δρr =δρ1 2δρ2+δρ3 (9) clear. Thischargeq andtwospin(q ,q )connectionscan − c l r only be easily established from CB approach presented We can rewrite H in Eqn.1 as: int in the next section. So we defer the discussion on the 1 1 1 classification of all the possible excitations to the next H = δρ Uδρ + δρ Uδρ + δρ Uδρ int 1 1 2 2 3 3 section. 2 2 2 Problems with MCF approach: +δρ1Vδρ2+δρ2Vδρ3+δρ1Wδρ3 AllthecriticismsonMCFapproachtoBLQHalsohold 1 1 1 = δρ V δρ + δρ V δρ + δρ V δρ +δρ V δρ (10) c c c l l l r r r c cr r toTLQH.Inthefollowing,welistjustoneswhicharethe 2 2 2 most relevant to this paper. where U = V ,V = V ,W = V and V = U + 4V + (a) It is easy to see that the spin wave dispersion in 11 12 13 c 3 9 2W,V = 1(U W),V = 1(U 2V +w),V = 1(V Eqn.8 remains linear ω vk even in the d 0 limit. 9 l 2 − r 3 2 − 3 6 cr −9 − ∼ → W). The stability of the system dictates V V V2 >0. Thiscontradictswiththewellestablishedfactthatinthe c r− cr In the long wavelength limit qd 1, we only keep the d 0limit,thelineardispersionrelationwillbereplaced ≪ → leading terms: V = 2π,V =2πd,V = 2πd,V = 2πd. by quadratic SU(3) Ferromagnetic spin-wave dispersion c q l r 9 cr − 9 relation ω k2 due to the enlarged SU(3) symmetry at Performing a singular gauge transformation: ∼ d 0. →(b) The broken symmetry in the ground state is not φa(~x)=ei d2x′arg(~x−~x′)ρ(~x′)ca(~x) (11) R obvious without resorting to the (111,111)wavefunction Eqn.28. The origin of the gapless mode is not clear. where ρ(~x)=c†1(~x)c1(~x)+c†2(~x)c2(~x)+c†3(~x)c3(~x) is the (c) The physical meaning of q,q is not clear. That total density of the tri-layer system. l r theyhavetobeintegerswasputinbyhandinanadhoc After absorbing the external gauge potential A~ into way. theChern-Simongaugepotential~a,wecantransformthe (d) The spin-charge connections in (q ,q ,q ) can not Hamiltonian Eqn.1 into the Lagrangian in the Coulomb c l r be extracted. gauge: (e) Even in the balanced case, there are two MCF cy- celvoetrr,otnhegraepissh¯onωlc∗yα,oαne=ch1a,r2geagtampeinanthfieelsdysttheemo.ryI.tiHsonwot- L=φ†a(∂τ −ia0)φa+φ†a(~x)(−i¯h∇~ 2−m¯h~a(~x))2φa(~x) knownhowtoreconcilethisdiscrepancywithinMCFap- +ia ρ¯+ 1 d2x′δρ (~x)V (~x ~x′)δρ (x~′) proach. 0 2Z c c − c (f) In the presence of interlayer tunneling, it is not 1 + d2x′δρ (~x)V (~x ~x′)δρ (x~′) knownhow to derivethe tunneling termina straightfor- 2Z l l − l ward way ( see section IV). 1 + d2x′δρ (~x)V (~x ~x′)δρ (x~′) In the following, we will show that the alternative CB 2Z r r − r approach not only achieve all the results, but also can i get rid of all these drawbacks. +Z d2x′δρc(~x)Vcr(~x−~x′)δρr(x~′)− 2πa0(∇×~a) (12) In the Coulomb gauge, integrating out a leads to the III. COMPOSITE BOSON APPROACH 0 constraint: ~a=2π(φ†φ ρ¯). ∇× a a− Itcanbe shownthatφ (~x)satisfiesallthebosoncom- a Compositebosonapproachwasfoundtobemuchmore mutation relations. We write the three bosons in terms effective than the MCF approach in BLQH. We expect of magnitude and phase it remains true in TLQH. In this section, we apply CB approach to study TLQH and find that it indeed avoids φ = ρ¯ +δρ eiθa (13) a a a all the problems suffered in MCF approach listed in the p last section. Instead of integrating out the charge de- The boson commutation relations imply that gree of freedoms which was done in the previous CB ap- [δρ (~x),φ (~x)] = i¯hδ δ(~x ~x′). For simplicity, in this a b ab − proach, we keep one charge sector and two spin sectors paper, we only consider the balanced case, so we set 4 ρ¯ = ρ¯. The imbalanced case can be discussed along which leads to the algebraic order: a similar lines developed in [12]. 1 We define the center of mass, left-moving and right- <ei(θc(~x)−θc(y~)) >= (18) x y 3 movingangleswhichareconjugateanglevariablestothe | − | three densities defined in Eqn.9 We coulddefine θ˜ =(θ +θ +θ )/3=θ /3,then the c 1 2 3 c θ =θ +θ +θ exponent will be 1/3. But when we consider topological c 1 2 3 θ =θ θ vortex excitations, then eiθ˜c may not be single valued. l 1 3 − Therefore, θ is more fundamental than θ˜. θ =θ 2θ +θ (14) c c r 1 2 3 − (b) Spin wave excitations: whichsatisfythecommutationrelations[δρα(~x),θβ(~x′)]= While in the spin sector,there are two neutral gapless Aαi¯hδαβδ(~x ~x′) where A=3,2,6 for α=c,l,r. modes: left-moving mode and right-moving mode [17]. − SubstitutingEqn13intoEqn.12,neglectingthemagni- Integrating out δρ leads to l tudefluctuationsinthespatialgradienttermwhichwere 1 1 ρ¯ shown to be irrelevant in BLQH in [12] and rewriting = ( ∂ θ )2+ ( θ )2 (19) l τ l l the spatial gradient term in terms of the three angles in L 2Vl(~q) 2 12m ∇ Eqn.14, we find: where the dispersion relation of spin wave can be ex- 1 ρ¯ 1 1 tracted: =iδρ ( ∂ θ a )+ [ θ ~a]2+ δρ V (~q)δρ c τ c 0 c c c c L 3 − 2m 3∇ − 2 2ρ¯ i ρ¯ 1 ω2 =[ V (~q)]q2 =v2q2 (20) + δρl∂τθl+ ( θl)2+ δρlVl(~q)δρl 3m l l 2 12m ∇ 2 + iδρr∂τθr+ ρ¯ ( θr)2+ 1δρrVr(~q)δρr Inthe long wave-lengthlimit qd≪1,the barespinwave 6 36m ∇ 2 velocity is: i +δρcVcr(~q)δρr a0( ~a) (15) ρ¯ 4πe2 2e2 d − 2π ∇× v2 = d= 2πρ¯( ) (21) l m 3ǫ 3mǫ l Inthe balancedcase,the symmetryis U(1) U(1) p c l × × U(1) Z where the first is a local gauge symmetry, Integrating out δρ leads to r × 2 r the second and the third are global symmetries and the 1 1 ρ¯ global Z2 symmetry is the exchange symmetry between r = ( ∂τθr)2+ ( θr)2 (22) layer 1 and layer 3. L 2Vr(~q) 6 36m ∇ (a) Off-diagonalAlgebraic order in the chargesector: where the dispersion relation of spin wave can be ex- Attemperaturesmuchlowerthanthevortexexcitation tracted: energy, we can neglect vortex configurations in Eqn.15 and only consider the low energy spin-wave excitation. 2ρ¯ ω2 =[ V (~q)]q2 =v2q2 (23) The charge sector ( θc mode ) and the two spin sectors ( m r r θ ,θ modes ) are essentially decoupled. It can be shown l r Inthe long wave-lengthlimit qd 1,the barespinwave that the second to the last term δρcVcr(~q)δρr in Eqn.15 ≪ velocity is: whichisthecouplingbetweenthechargesectorandright- moving sector is irrelevantin the long wave-lengthlimit. ρ¯ 4πe2 Therefore,thechargesectorisessentiallythesameasthe v2 = d=v2/3 (24) r m 9ǫ l CSGL action in BLQH. Using the constraint a = 2πδρc, t q neglecting vortex excitations in the groundstate and in- Bothspinwavevelocitiesshouldincreaseasthesquare tegrating out δρc leads to the effective action of θc: root of the separation d when d < dc/2, their ratio v /v √3. At d = 0, v = v = 0. This is expected, 1 1 ω2+ω2 l r ∼ l r = ( )2θ ( ~q, ω)[ q~ ]θ (~q,ω) (16) because at d = 0 the U(1)l U(1)r Z2 symmetry is Lc 2 × 3 c − − V (q)+ 4π2ρ¯ 1 c enlarged to SU(3) , the spi×n wave o×f SU(3) isotropic c m q2 G ferromagnet ω k2. whereωq~2 =ωc2+mρ¯q2Vc(q)andωc = 2mπρ¯ isthecyclotron We have also∼treated the coupling term between the frequency. charge and right-moving sector δρ V (~q)δρ in Eqn.15 c cr r From Eqn.16, we can find the equal time correlatorof exactly and found that in the ~q,ω 0 limit, the mixed θc: propagator takes the form: → ∞ dω <θ ( ~q)θ (~q)> = <θ ( ~q, ω)θ (~q,ω)> 9(m)2V ω2 c − c Z 2π c − − c <θ ( ~q, ω)θ (~q,ω)>= 2 πρ¯ cr ω2 <θ θ > −∞ c − − r ω2+v2q2 ∼ r r 2π 1 r =3 +O( ) (17) × q2 q (25) 5 which is dictated by < θ θ > instead of < θ θ >. Be- proach with Eqn.7 derived in MCF approach. Just like r r c c cause of the ω2 prefactor, we can ignore <θ θ > in the inBLQH,the threetopologicalvortexcurrentsaremiss- c r low energy limit. inginEqn.7,whileχ ,χ ,χ ,δχtermsareextraspurious c l r We also found that Eqns.16, 17, 18 and Eqns.22,23,24 terms which break SU(3) symmetry even in the d 0 → remaintrueinthe long-wavelengthandlowenergylimit, limit. If we drop all the χ terms, use bare mass and therefore confirm that δρ V (~q)δρ is indeed irrelevant addthe three vortexcurrents terms in Eqn.7, the result- c cr r in the limit. ing effective action will be identical to the dual action As shown in BLQH in [12], the functional form of the Eqn.26. Neglecting the vortex currents, we can identify spin sector in the CB theory is the same as EPQFM. the two spin wave velocities easily from the dual action Although there are no detailed microscopic calculations which are the same as found previously from Eqn.15. It inTLQHyetexceptthe preliminaryworkin[6,16],from isalsoeasytoshowthatthelastterminEqn.26whichis theinsightsgainedinBLQH,wecanclaimthatthefunc- the couplingbetweenthechargeandright-movingsector tional form of the spin sector in Eqn.15 ( or Eqn.33 ) is irrelevant in the low energy limit. We conclude that achievedinthe CBtheoryiscorrect. Again,itis hardto Eqns.15 and its dual action Eqn. 26 are the correct and incorporate the LLL projection in this CB approach, So complete effective actions. some parameters can not be evaluated within this ap- (d) Topological excitations: proach. For example, the two spin stiffnesses for the Any topological excitations are characterizedby three left and right-moving sectors in Eqn.15 ( or or Eqn.33 winding numbers ∆θ = 2πm ,∆θ = 2πm ,∆θ = 1 1 2 2 3 ) should be completely determined by the Coulomb in- 2πm , or equivalently, ∆θ = 2π(m + m + m ) = 3 c 1 2 3 teractions instead of being dependent of the band mass, 2πm ,∆θ =2π(m m )=2πm ,∆θ =2π(m 2m + c l 1 3 l r 1 2 − − theymaybe estimatedbythe microscopicLLL+HFap- m )=2πm . Itisimportanttostressthatthethreefun- 3 r proach. However,theratioofthetwovelocitiesinEqn.24 damental angles are θ ,θ ,θ instead of θ ,θ ,θ . There- 1 2 3 c l r isindependentofthebandmass,itisinterestingtoseeif fore, m ,m ,m are three independent integers, while 1 2 3 thisratio√3remainscorrectinthemicroscopicLLL+HF m ,m ,m are not. c l r estimation. JustlikeinBLQH,wecansimplytakeρ ,V Therearefollowing6kindsoffundamentaltopological sl l and ρ ,V in Eqn.15 as phenomilogical parameters to excitations: ∆φ = 2π or ∆φ = 2π or ∆φ = 2π, sr r 1 2 3 ± ± ± be fitted into the LLL + HF estimations. However, just namely (m ,m ,m ) = ( 1,0,0),(0, 1,0),(0,0, 1). 1 2 3 ± ± ± like in BLQH [10,12,19], because the qualitative correct Theycorrespondtoinsertingonefluxquantuminlayer1 ground state wavefunction in TLQH is still unknown, it or2or3inthesameoroppositedirectionastheexternal is still difficult to estimate these parameters with con- magnetic field. Let’s classify the topological excitations trolledaccuracyevenin the LLL+HF approach. We can intermsof(q,m ,m )where q isthe fractionalchargeof l r simply fit these parameters into experimental data. the topological excitations in the following table. (c) Dual action: ( 1,0,0) (0, 1,0) (0,0, 1) Just like in BLQH discussed in [12], we can per- ± ± ± m 1 1 1 form a duality transformation on Eqn.15 to obtain a c ± ± ± m 1 0 1 dual action in terms of three vortex currents Jv;c,l,r = l ± ∓ µ m 1 2 1 1 ǫ ∂ ∂ θ and three dual gauge fields bc,bl,br: r ± ∓ ± 2π µνλ ν λ c,l,r µ µ µ q 1/3 1/3 1/3 ± ± ± 2π = iπbcǫ ∂ bc iAc ǫ ∂ bc +i bcJvc Table1: ThefractionalchargeinbalancedTLQH Ld − µ µνλ ν λ− sµ µνλ ν λ 3 µ µ The fractional charges in the table were determined m 1 + 2ρ¯f(∂αbc0−∂0bcα)2+ 2(∇×~bc)Vc(~q)(∇×~bc) from the constraint ∇×~a = 2πδρc and the finiteness of the energy in the charge sector: iAl ǫ ∂ bl +iπblJvl − sµ µνλ ν λ µ µ 1 1 1 1 + 34mρ¯(∂αbl0−∂0blα)2+ 12(∇×~bl)Vl(~q)(∇×~bl) q = 2π I ~a·d~l= 2π × 3I ∇θc·d~l = 3mc (27) π iAr ǫ ∂ br +i brJvr There are the following two possible bound states: − sµ µνλ ν λ 3 µ µ (1) Three Charge neutral two-body bound states: m 1 + (∂ br ∂ br)2+ ( ~br)V (~q)( ~br) 4ρ¯ α 0− 0 α 2 ∇× r ∇× Layer 1: ( 1/3, 1, 1) with energy E1 = Ec+ + +( ~bc)Vcr(~q)( ~br) (26) • Ec−− 9eR2 +±2π(ρ±sl+±ρsr)lnRRc when R≫Rc ≫l. ∇× ∇× Layer 2: ( 1/3,0, 2) with energy E = E + where the three source gauge fields are Acsµ = A1sµ + • E e2 +±2π(4ρ ∓)ln R. 2 c+ A +A ,Al = A A ,Ar =A 2A + c−− 9R sr Rc 2sµ 3sµ sµ 1sµ− 3sµ sµ 1sµ− 2sµ A . Layer 3: ( 1/3, 1, 1) with energy E = E 3sµ 3 1 • ± ∓ ± We cancompare this dual action derivedfrom CB ap- which is dictated by Z symmetry. 2 6 The corresponding 2-body Kosterlize-Thouless (KT) implies m = m = m = m and m = 3m. Eqn.27 im- 1 2 3 c transitions are k T2b =k T2b = π(ρ +ρ ) which plies the charge q = 1m = m must be an integer. This B KT1 B KT3 2 sl sr 3 c is dictated by Z symmetry and k T2b =2πρ above proof rigorouslyrules out the possibility of the existence 2 B KT2 sr which the QP and QH pairs are liberated into free ones. of deconfined fractional charges. We conclude that any If setting ρ = 3ρ , then there is only one 2-body KT deconfined charge must have an integral charge. m = 1 sl sr temperature k T2b =2πρ = πρ¯. As explained previ- corresponds to inserting one flux quantum through all B KT sr 9m ously,microscopiccalculationsintheLLL[22]areneeded the three layers which is conventional charge 1 excita- to estimate the two spin stiffnesses ρ and ρ . In gen- tion. Splitting the whole flux quantum into three fluxes sl sr eral, there should be two 2-body KT transition temper- penetrating the three layers at three different positions atures k T2b = k T2b = k T2b . The spin wave will turn into the three-body bound state with the same B KT1 B KT3 6 B KT2 excitations will turn into these Chargeneutral two-body chargeshown in Fig.1. It is still not known if the energy bound states at large wavevector ( or short distance ). ofthis conventionalcharge1 excitationis lowerthan the The two-body bound state behaves as a boson. three body bound state. (1) Two Charge 1 three-body bound states: ± The three-body bound states consists of three quasi- particles ( 1/3, 1, 1),( 1/3,0, 2),( 1/3, 1, 1) ± ± ± ± ∓ ± ∓ ± located at the three corners of a triangle ( Fig.1). IV. BROKEN SYMMETRY (1/3,1,1) ThebrokensymmetryandassociatedGoldstonemodes can be easily seen from wavefunction approach. In the R R 12 first quantization, in the d 0 limit, the ground state 13 → trial wavefunction is the (111,111)state: N1 N2 N1 N3 N2 N3 (1/3,0,−2) R (1/3,−1,1) Ψ = (u v ) (u w ) (v w ) 23 111,111 i− j i− j i− j YY YY YY i=1j=1 i=1j=1 i=1j=1 Fig 1: The lowest energy charged excitation is a three-body N1 N2 N3 bound state of three ±1/3 charged quasi-particles sitting on the (ui uj) (vi vj) (wi wj) (28) × − − − threecornersofatrianglewithR12=R23dictatedbytheZ2sym- iY<j Yi<j iY<j metry. Thethreequantumnumbers(q,ml,mr)areenclosedinthe parenthesis where u,v,w are the coordinates in layer 1, 2 and 3 re- Its energy E = 3E + e2( 1 + 1 + 1 ) + spectively. 3b c± 9 R12 R23 R13 2π[2ρ lnR12+(ρ ρ )lnR13+2ρ lnR23]whenR In the second quantization, a trial wavefunction was sr Rc sl− sr Rc sr Rc ≫ R l. Minimizing E with respect to R ,R ,R written in [16]: c 3b 12 13 23 lead≫s to two optimal separations: R = R = e2 o12 o23 36πρsr which is dictated by the Z symmetry and R = 2 o13 M−1 p18aπr(tρieclr2l−esρsar)r.e Ilfocsaettetidngonρstlh=e 3coρrsnr,ertsheonf taheisothscreeleesquwaistih- |Ψ> = mY=0 √13(eiθuCm†,1+Cm†,2+eiθdCm†,3)|0> lengthR = e2 ,thecorresponding3-bodyKosterlize- M−1 Tbohuonudlessstoa(tKeT3k6)πρTtsrr3abnsi=tiokn iTs2tbhe=sa2mπeρas=theπρ¯tw. oT-bhoedsye = mY=0 √13(1+eiθuCm†,1Cm,2+eiθdCm†,3Cm,2) B KT B KT sr 9m chargedexcitationsbehaveas fermions andarethe main M−1 C† 0> (29) sources of dissipations in transport experiments. In gen- × m,2| Y eral, T3b should be different from the two 2-body KT m=0 KT transitiontemperatures. Again,microscopiccalculations ofthe twospin stiffnesses ρsl andρsr inthe LLL [22]are where θu = θ1 θ2,θd = θ3 θ2 and M = N is the − − needed to determine the three KT transition tempera- angular momentum quantum number corresponding to tures. the edge. We caninterpret θ mode in Eqn.29as a ”up” u Let’s look at the interesting possibility of deconfined pairingbetweenanelectroninlayer1 anda holeinlayer ( or free ) 1/2 charged excitations. Because any excita- 2 which leads to a θ molecule, similarly, θ mode as a ” u d tions with non-vanishing m or m will be confined, so down” pairing betweenanelectronin layer3 anda hole l r anydeconfinedexcitationsmusthavem =m =0which in layer 2 which leads to a θ molecule ( Fig. 2). l r d 7 iseasiertoperformtransportexperimentsinthecounter- θ flow channel than in the right-moving channel. u I θ I u θ d θ I d I (3a) Fig 2: Two pairing modes of TLQH. Solid dots are electrons, I opendotsareholes. θ I u 2I Note that when θu = θd = 0, Ψ > is the ground θ I | d state G >, it can be shown [19] that its projection on | I (N ,N ,N ) sector is (111,111)wavefunctionin Eqn.28. 1 2 3 (3b) Note that Eqn.28 is correct only in d 0 limit where → the symmetry is enlarged to SU(3). In the SU(3) sym- Fig3: (3a)Left-movingsuperfluidchannelwhichisthecounter- metric case, the pairing between any two layers of the flowchannel (3b)Right-movingsuperfluidchannel [17] three layers is equally likely. However, at any finite d, Eqn.28 and Eqn.29 may even not be qualitatively cor- rect [19], because the SU(3) symmetry is reduced to V. CORRELATED INTERLAYER TUNNELINGS U(1) U(1) U(1) Z symmetry ( where the three 2 × × × U(1) symmetry correspond to cα(~x) eiθαcα(~x),α = → 1,2,3,c c )Thepairingsbetweenanytwolayersout So far, we assume that the tunnelings are tuned to 1 3 ↔ of the three layersarenot equivalentany more. The two be zero. In this section, we will outline several salient lowest energy pairing modes are shown in Fig. 2. features inthe tunnelings of TLQH.We assumethe tun- SU(3) has 8 generatorswhich can be written in terms neling amplitude from layer 1 to layer 2 is the same as of c : that of from layer 2 to layer 3. The tunneling term is: α Sl(~x)=c†1(~x)c1(~x)−c†3(~x)c3(~x) Ht =tc†1c2+h.c.+tc†2c3+h.c.=tφ†1φ2+h.c.+tφ†2φ3+h.c. Sr(~x)=c†1(~x)c1(~x)+c†3(~x)c3(~x)−2c†2(~x)c2(~x) =tρ¯(cosθ +cosθ )=2tρ¯cosθl cosθr (31) S (~x)=c†(~x)c (~x), S (~x)=S† (~x)=c†(~x)c (~x) u d 2 2 12 1 2 21 12 2 1 S23(~x)=c†2(~x)c3(~x), S32(~x)=S2†3(~x)=c†3(~x)c2(~x) where θu =θ1 θ2,θd =θ3 θ2 − − S13(~x)=c†1(~x)c3(~x), S31(~x)=S1†3(~x)=c†3(~x)c1(~x) (30) Integrating out the gapped charge sector θc leads to the effective tunneling action: where S ,S are the two ( left and right ) generators in l r Cartansubalgebra(NotethatthealgebraisnotSU(2)in 1 1 = [χ (∂ θ )2+ρ ( θ )2]+ [χ (∂ θ )2+ρ ( θ )2] spin j = 1 representation which is also 3 dimensional as L 2 l τ l sl ∇ l 2 r τ r sr ∇ r usedin[6]). InthebalancedcaseN1 =N2 =N3 =N/3, θl θr +2tρ¯cos cos (32) wehave<S >=<S >=0. ThegroundstateinEqn.28 l r 2 2 breaks twoofthethreeU(1)symmetries,thereforethere aretwoGoldstonemodes inthe correspondingtwoorder where, as noted previously, the two spin susceptibilities parameters <S12 >= N3eiθu,<S23 >= N3e−iθd. χl,χr and the two spin-stiffnesses ρsl,ρsr are phenomi- When θ molecule and θ molecule in Fig.2 move in logical parameters to be fitted into the LLL+HF calcu- u d theoppositedirection,thecurrentsinlayer2canceleach lations or experimental data. other, it leads to the left-moving superfluid channel θ ( Unfortunately, the above action is not very useful l Fig. 3a). Whilewhenθ moleculeandθ moleculemove for any practical calculations, because the two angles u d inthesamedirection,thecurrentsinlayer2addandflow θ =θ θ =θ θ ,θ =θ +θ =θ 2θ +θ arenot l u d 1 3 r u d 1 2 3 − − − in the opposite direction to the currents in layer 1 and independent angles when considering the vortex excita- layer 3, it leads to the right-moving superfluid channel tions. We can see this fact by noting that θ +θ = 2θ l r u θ ( Fig. 3b ). In BLQH, there is only one superfluid or θ θ = 2θ has to be twice an angle which is a r l r d − − channel which is the counter-flowchannel [5]. In TLQH, constraint between θ and θ . While θ and θ are two l u u d there are two superfluid channels which are left-moving independent angles. So it is much more useful to write and right-moving channels, the left-moving channel cor- the above tunneling action in terms of the two indepen- responds to the counter-flow channel [17]. Obviously, it dent angles θ ,θ : u d 8 = 1(χ +χ )(∂ θ )2+ 1(ρ +ρ )( θ )2+tρ¯cosθ VI. CONCLUSIONS: l r τ u sl sr u u L 2 2 ∇ 1 1 + (χ +χ )(∂ θ )2+ (ρ +ρ )( θ )2+tρ¯cosθ l r τ d sl sr d d 2 2 ∇ +(χl χr)∂τθu∂τθd+(ρsl ρsr) θu θd (33) In this paper, we study the interlayer coherent incom- − − ∇ ·∇ pressible phase in spin polarized Trilayer Quantum Hall systems at total filling factor ν = 1 from three ap- where there is a Z symmetry under θ θ . The T 2 u d ↔ proaches: MCF, CB and wavefunction approach. Just tunneling currents in three layers can be expressed in like in TLQH, CB approach is superior than MCF ap- terms of the two independent angles: I =tρ¯sinθ ,I = 1 u 2 proach. The Hall and Hall drag resistivities are found tρ¯(sinθ +sinθ ),I = tρ¯sinθ . They satisfy the cur- u d 3 d − to be quantized at h/e2. Two neutral gapless modes rent conservation I +I +I =0. 1 2 3 which are left and right moving modes with linear dis- From Eqn.33, we can see there is a coupling between persionrelationsv =ω k areidentified andthe ratio θ andθ . Iftherewerenotthiscoupling,wewouldhave l/r l/r u d of the two velocities is close to √3. The novel excita- twoindependent BLQHtunneling actions. Itis this cou- tion spectra are classified into two classes: (1) Charge pling which lead to the correlated tunnelings of TLQH. neutral bosonic two-body bound states. (3) Charge 1 Forexample,thecurrentoutoflayer1I =tρ¯sinθ only 1 u ± fermionic three-body bound states. In general, there are depends on θ explicitly, while θ couples to θ . It was u u d two 2-body KT transition temperatures and one 3-body shown that the single gapless mode in BLQH leads to a KT transition temperature. Microscopic calculations in sharpzero-biaspeakintheinterlayertunnelingofBLQH, the LLL [22] are needed to roughly estimate the three the experimentally observed dissipations can only come KT transitiontemperatures. The Charge 1 three-body from external sources such as disorder [21]. If the cou- ± boundstatesshowninFig. 1maybethemaindissipation pling in TLQH even in the absence of external sources source of the interlayer tunneling. The broken symme- will lead to a broad zero-bias tunneling peak in TLQH try in terms of SU(3) algebra is studied. The effective will be investigated in [22]. action of the two coupled Goldstone modes are derived. In the presence of in-plane magnetic field B = || The excitonic structure in TLQH was shown in Fig. 2. (B ,B ), the gauge invariance dictates the tunneling x y In BLQH, there is only one superfluid channel which is term: thecounter-flowchannel[5]. InTLQH,therearetwosu- perfluidchannelswhichareleft-movingandright-moving Ht =tρ¯(cos(θu Q x)+cos(θd+Q x)) (34) channels ( Fig.3 ), the left-moving channel corresponds − · · to the counter-flow channel ( Fig. 3a ). Obviously, it is where Q =( 2πdBy,2πdBx). easier to perform transport experiments in the counter- α − φ0 φ0 flow channel than in the right-moving channel. In the InBLQH,itwasfoundthatthein-planefieldwillsplit presence of interlayer tunnelings, the coupling between the zero-bias peak, the splitting gives a direct measure- the two Goldstone modes may lead to the broadening ment of the spin-wave velocity of the single Goldstone of the zero-bias tunneling peak, in contrast to the zero- mode in BLQH [21]. However, in TLQH, there are two biaspeakinBLQH.Theprecisenatureofthebroadening coupled Goldstone modes with two different velocities andseveralother possible new phenomena unique to the v ,v . In[22],wewillstudyhowthezero-biaspeakshifts l r TLQH in the presence of in-plane magnetic field will be andhowtorelatetheshifttothetwospinwavevelocities studied in a forthcoming publication. or their ratio in the presence of in-plane magnetic field. We expect that the effective action of the two coupled We can also see that if there were not the coupling Goldstone modes and the qualitative physical pictures between θ and θ , we would have two independent u d of TLQH achieved from CB approach in this paper are Pokrovsky-Talapov(PT) models. In BLQH which is de- correct. However, the parameters in the effective action scribed by a single PT model, it was found that when needtobeestimatedfrommicroscopiccalculationsinthe the applied in-plane magnetic field is larger than a criti- LLL [22]. It is interesting to see if the ratio v /v √3 calfield B >B∗, there is a phase transitionfrom a com- l r ∼ || still holds in the microscopic calculations. As stressed mensuratestatetoanincommensuratestate(C-IC)with in [10,19] for BLQH, the trial wavefunction Eqn.28 and broken translational symmetry. When B > B∗, there || Eqn.29 may not even be qualitatively correct in TLQH, is a finite temperature KT transition which restores the so it is still difficult to estimate the ratio with controlled translationsymmetrybymeansofdislocationsinthedo- accuracyevenintheLLL+HFapproach. Eventually,the main wall structure in the incommensurate phase [9,11]. ratio need to be measured by experiments. In TLQH which is described by the above two coupled PT model [20], several kinds of C-IC transitions and as- I thank J. K. Jain for helpful discussions. sociatedseveralkindsofKTtransitionsareexpectedand will be studied in a separate publication [22]. 9 [1] B.I.Halperin,Helv.Phys.Acta56, 75(1983); Surf.Sci. [14] See the review by G. Murthy and R. Shankar, Rev. of 305, 1 (1994) Mod. Phys.75, 1101, 2003. [2] J. P. Eisenstein, L. N. Pfeiffer and K. W. West, Phys. [15] J. Jo, Y. W. Suen, L. W. Engel, M. B. Santos, and M. Rev. Lett. 69, 3804 (1992); Song He, P. M. Platzman Shayegan,Phys.Rev.B46, 9776-9779 (1992); T.S.Lay, and B. I. Halperin, Phys. Rev.Lett. 71, 777 (1993). X.Ying,andM.Shayegan,Phys.Rev.B52,R5511-5514 [3] I. B. Spielman et al, Phys. Rev. Lett. 84, 5808 (2000). (1995). ibid, 87, 036803 (2001). [16] C.B.HannaandA.H.MacDonald,Phys.Rev.B53(23): [4] M. Kellogg, et al, Phys.Rev. Lett.88, 126804 (2002). 15981 (1996). [5] J.P.Eisenstein,A.H.MacDonald,cond-mat/0404113;M. [17] The ”left” and ”right ” channels are used just to distin- Kelloggetal,cond-mat/0401521;E.Tutuc,M.Shayegan, guish thetwogapless channels.Theyhavenothingtodo D.A.Huse, cond-mat/0402186. with the movingdirections in thereal space. [6] H.Fertig, Phys.Rev.B 40, 1087 (1989). [18] A.LopezandE.Fradkin,Phys.Rev.B.44,5246(1991). [7] X.G.WenandA.Zee,Phys.Rev.Lett.69,1811(1992). [19] Gun Sang Jeon and Jinwu Ye, cond-mat/0407258. To [8] Z. F. Ezawa and A. Iwazaki, Phys. Rev. B. 48, 15189 appear in Phys. Rev.B. (1993). [20] This relation resembles that between Ising model and [9] KunYang et al, Phys.Rev. Lett.72, 732 (1994). Ashkin-Teller model which is a model of two coupled [10] K.Moon et al, Phys. Rev.B 51, 5138 (1995). Ising models. [11] For reviews of bilayer quantum Hall systems, see S. M. [21] L.BalentsandL.Radzihovsky,Phys.Rev.Lett.86,1825 GirvinandA.H.Macdonald,inPerspectivesinQuantum (2001). A. Stern,S.M. Girvin, A.H.Macdonald and N. Hall Effects, edited by S. Das Sarma and Aron Pinczuk Ma, ibid86, 1829(2001).M.FoglerandF.Wilczek,ibid ( Wiley, New York,1997). 86, 1833 (2001). [12] Jinwu Ye,cond-mat/0310512 [22] Jinwu Ye, unpublished. [13] Jinwu Ye,cond-mat/0407088 10